Escape velocity is the speed an unpowered object needs to break free of a planet's gravity for good — about 11.2 km per second from Earth's surface.
Throw a stone up and it comes back. Throw it faster and it goes higher before returning. Escape velocity is the speed at which, ignoring air, it would never come back. It would keep slowing down forever but never stop and fall.
NASA's Earth Fact Sheet gives Earth's escape velocity as 11.186 km/s. For the Moon it is only 2.38 km/s, because the Moon is much smaller and lighter.
A satellite in low orbit needs about 7.7 km/s — enough to keep falling around Earth, not enough to leave. Escape velocity is always √2, about 1.41, times the circular orbit speed at the same height.
So from the height of the ISS, about 400 km up, escape would need about 10.9 km/s instead of the 7.7 km/s the station actually has.
Chandrayaan-3 did not leave the launch pad at escape velocity. ISRO's LVM3 rocket, launched from Sriharikota, placed it in an elliptic parking orbit of about 170 × 36,500 km. That is still a closed orbit around Earth.
From there, the mission's propulsion module carried the lander towards the Moon and into a 100 km lunar orbit.
Escape velocity applies to something that is coasting with no engine running. A rocket that keeps firing can climb slowly and still leave Earth. The idea is about energy: the object must carry enough kinetic energy to pay for the whole climb out of Earth's gravity.
The Velocity card reads about 7.66 km/s; to escape Earth from that height the station would need about 10.9 km/s. Open the ISS tracker
Escape is an energy condition. An object of mass m at distance r has kinetic energy ½mv² and gravitational potential energy −GMm/r. It escapes if its total energy is zero or more:
½ v² − GM / r ≥ 0 → v_esc = √(2GM / r)
Compare with circular speed v_c = √(GM / r): v_esc = √2 × v_c.
Earth's surface. r = 6,378 km: v_esc = √(2 × 398,600 ÷ 6,378) = √125.0 ≈ 11.18 km/s, matching the fact sheet's 11.186 km/s.
ISS height. r = 6,778 km: v_esc = √(2 × 58.81) = √117.6 ≈ 10.85 km/s. The station's circular speed is 7.67 km/s, so it would need about 3.2 km/s more.
The Moon. GM = 4,900 km³/s², radius 1,737.4 km: v_esc = √(9,800 ÷ 1,737.4) = √5.64 ≈ 2.38 km/s. A lander lifting off the Moon has a far easier job than a rocket leaving Earth.
Was Chandrayaan-3's parking orbit an escape path? At its perigee (r = 6,548 km) the vis-viva equation gives a speed of about 10.28 km/s. Escape speed at that distance is √(2 × 398,600 ÷ 6,548) = √121.75 ≈ 11.03 km/s. The spacecraft was about 0.75 km/s short, so it stayed bound to Earth until it was sent onward.
One caution: the Moon itself sits inside Earth's gravitational hold. Reaching the Moon does not require full escape from Earth; it requires an orbit whose high point reaches the Moon's distance (about 384,400 km), where the Moon's gravity can capture the spacecraft.
v_esc = √(2GM/R). It does not depend on the mass of the escaping object.
Escape velocity is √2 (about 1.41) times the circular orbital speed at that distance.
The Moon has much less mass. Its GM is about 1/81 of Earth's, and even with its smaller radius the result is a much lower escape speed.